Department of Informatics, University of Oslo, P. O. Box 1080, NO-0316 Oslo, Norway.
J Acoust Soc Am. 2011 Oct;130(4):2195-202. doi: 10.1121/1.3631626.
This work presents a lossy partial differential acoustic wave equation including fractional derivative terms. It is derived from first principles of physics (mass and momentum conservation) and an equation of state given by the fractional Zener stress-strain constitutive relation. For a derivative order α in the fractional Zener relation, the resulting absorption α(k) obeys frequency power-laws as α(k) ∝ ω(1+α) in a low-frequency regime, α(k) ∝ ω(1-α/2) in an intermediate-frequency regime, and α(k) ∝ ω(1-α) in a high-frequency regime. The value α=1 corresponds to the case of a single relaxation process. The wave equation is causal for all frequencies. In addition the sound speed does not diverge as the frequency approaches infinity. This is an improvement over a previously published wave equation building on the fractional Kelvin-Voigt constitutive relation.
本文提出了一个包含分数导数项的有损偏微分声波方程。它是从物理的第一性原理（质量和动量守恒）和由分数泽纳应力-应变本构关系给出的状态方程推导出来的。对于分数泽纳关系中的导数阶数α，所得到的吸收α(k)在低频区遵循频率幂律关系，即α(k)∝ω(1+α)，在中频区遵循α(k)∝ω(1-α/2)，在高频区遵循α(k)∝ω(1-α)。当α=1 时，对应于单个弛豫过程的情况。对于所有频率，波动方程都是因果的。此外，当频率接近无穷大时，声速不会发散。这比基于分数 Kelvin-Voigt 本构关系的先前发布的波动方程有所改进。
